A [12,12,18] Hurwitz sum-of-squares formula exists over fields with -1 a square but not over formally real fields, disproving Shapiro's conjecture.
Albert,Quadratic forms permitting composition, Ann
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The Hurwitz sum-of-squares problem depends on the base field
A [12,12,18] Hurwitz sum-of-squares formula exists over fields with -1 a square but not over formally real fields, disproving Shapiro's conjecture.